Proving the existence of a solution of \(F=0\)  using Theorem REF  is often called the radii polynomial approach (see e.g. [1]}, [2]}). In practice, this approach consists of considering a finite dimensional projection of (REF ), computing an approximate solution \(\bar{a}\)  (i.e. such that \(F(\bar{a}) \approx 0\) ), considering an approximate derivative \(A^\dag \)  of the derivative \(DF(\bar{a})\)  and an approximate inverse \(A\)  of \(DF(\bar{a})\) . Once the numerical approximation \(\bar{a}\)  and the linear operators \(A\)  and \(A^\dag \)  are obtained, formulas for the bounds \(Y_0\) , \(Z_0\) , \(Z_1\)  and \(Z_2(r)\)  are derived analytically and finally implemented in a computer-program using interval arithmetic (see [3]}). The final step is to find (if possible) a radius \(r_0>0\)  for which \(p(r_0)<0\) . In case such an \(r_0\)  exists, it naturally provides a \(C^0\)  bound for the error between the approximate solution \((\bar{u}(x),\bar{v}(x))\)  and the exact solution \((\tilde{u}(x),\tilde{v}(x))\) , which have Fourier coefficients \(\bar{a}\)  and \(\tilde{a}\) , respectively, see (REF ).
The following remark makes this statement explicit.
